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Mathematics > Combinatorics

Title:Ideal decompositions and computation of tensor normal forms

Abstract: Symmetry properties of r-times covariant tensors T can be described by
certain linear subspaces W of the group ring K[S_r] of a symmetric group S_r.
If for a class of tensors T such a W is known, the elements of the orthogonal
subspace W^{\bot} of W within the dual space of K[S_r] yield linear identities
needed for a treatment of the term combination problem for the coordinates of
the T. We give the structure of these W for every situation which appears in
symbolic tensor calculations by computer. Characterizing idempotents of such W
can be determined by means of an ideal decomposition algorithm which works in
every semisimple ring up to an isomorphism. Furthermore, we use tools such as
the Littlewood-Richardson rule, plethysms and discrete Fourier transforms for
S_r to increase the efficience of calculations. All described methods were
implemented in a Mathematica package called PERMS.